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In mathematics, many notations have been invented for the conciserepresentation of mathematical formulae. Tensor index notation is one of suchnotations and has been playing a crucial role in describing formulae inmathematical physics. This paper shows a programming language that can dealwith symbolical tensor indices by introducing a set of tensor index rules thatis compatible with two types of parameters, i.e., scalar and tensor parameters.When a tensor parameter obtains a tensor as an argument, the function treatsthe tensor argument as a whole. In contrast, when a scalar parameter obtains atensor as an argument, the function is applied to each component of the tensor.On a language with scalar and tensor parameters, we can design a set of indexreduction rules that allows users to use tensor index notation for arbitraryuser-defined functions without requiring additional description. Furthermore,we can also design index completion rules that allow users to define theoperators concisely for differential forms such as the wedge product, exteriorderivative, and Hodge star operator. In our proposal, all these tensoroperators are user-defined functions and can be passed as arguments ofhigh-order functions.

Symbolical Index Reduction and Completion Rules for Importing Tensor Index Notation into Programming Languages